Amalgamations of the Painlevé Equations

We present new hierarchies of nonlinear ordinary differential equations (ODEs) that are generalizations of the Painlevé equations. These hierarchies contain the Painlevé equations as special cases. We emphasize the sixth-order ODEs. Special solutions for one of them are expressed via the general solutions of the P ₁ and P ₂ equations and special cases of the P ₃ and P ₅ equations. Four of the six Painlevé equations can be considered special cases of these sixth-order ODEs. We give linear representations for solving the Cauchy problems for the hierarchy equations using the inverse monodromy transform.     

Invariant Simultaneous Solutions of Evolution Equations of Integrable Hierarchies

We construct classical point symmetry groups for joint pairs of evolution equations (systems of equations) of integrable hierarchies related to the auxiliary equation of the method of the inverse problem of second order. For the two cases: the hierarchy of Korteweg--de Vries (KdV) equations and of the systems of Kaup equations, we construct simultaneous solutions invariant with respect to the symmetry group. The problem of the construction of these solutions can be reduced, respectively, to the first and second Painlevé equations depending on a parameter. The Painlevé equations are supplemented by the linear evolution equations defining the deformation of the solution of the corresponding Painlevé equation.     

Movable Singularities of Solutions of Difference Equations in Relation to Solvability and a Study of a Superstable Fixed Point

We review applications of exponential asymptotics and analyzable function theory to difference equations in defining an analogue of the Painlevé property for them, and we sketch the conclusions about the solvability properties of first-order autonomous difference equations. If the Painlevé property is present, the equations are explicitly solvable; otherwise, under additional assumptions, the integrals of motion develop singularity barriers. We apply the method to the logistic map x _n+1=ax _n (1–x _n ), where we find that the only cases with the Painlevé property are a=–2,0,2, and 4, for which explicit solutions indeed exist; otherwise, an associated conjugation map develops singularity barriers.     

Coulomb Gas Representation for Rational Solutions of the Painlevé Equations

We consider rational solutions for a number of dynamic systems of the type of the nonlinear Schrödinger equation, in particular, the Levi system. We derive the equations for the dynamics of poles and Bäcklund transformations for these solutions. We show that these solutions can be reduced to rational solutions of the Painlevé IV equation, with the equations for the pole dynamics becoming the stationary equations for the two-dimensional Coulomb gas in a parabolic potential. The corresponding Coulomb systems are derived for the Painlevé II–VI equations. Using the Hamiltonian formalism, we construct the spin representation of the Painlevé equations.     

Lie point symmetries, partial Noether operators and first integrals of the Painlevé-Gambier equations

We utilize the Lie-Tressé linearization method to obtain linearizing point transformations of certain autonomous nonlinear second-order ordinary differential equations contained in the Painlevé-Gambier classification. These point transformations are constructed using the Lie point symmetry generators admitted by the underlying Painlevé-Gambier equations. It is also shown that those Painlevé-Gambier equations which have a few Lie point symmetries and hence are not linearizable by this method can be integrated by a quadrature. Moreover, by making use of the partial Lagrangian approach we obtain time dependent and time independent first integrals for these Painlevé-Gambier equations which have not been reported in the earlier literature. A comparison of the results obtained in this paper is made with the ones obtained using the generalized Sundman linearization method.     

Folding transformations of the Painlevé equations

New symmetries of the Painlevé differential equations, called folding transformations, are determined. These transformations are not birational but algebraic transformations of degree 2, 3, or 4. These are associated with quotients of the spaces of initial conditions of each Painlevé equation. We make the complete list of such transformations up to birational symmetries. We also discuss correspondences of special solutions of Painlevé equations.Acknowledgement The authors wish to thank Prof. Yosuke Ohyama, Prof. Shun Shimomura, and Dr. Yoshikatsu Sasaki for valuable discussions.     

Painlevé Hierarchies and the Painlevé Test

In a series of recent papers, we derived several new hierarchies of higher-order analogues of the six Painlevé equations. Here we consider one particular example of such a hierarchy, namely, a recently derived fourth Painlevé hierarchy. We use this hierarchy to illustrate how knowing the Hamiltonian structures and Miura maps can allow finding first integrals of the ordinary differential equations derived. We also consider the implications of the second member of this hierarchy for the Painlevé test. In particular, we find that the Ablowitz–Ramani–Segur algorithm cannot be applied to this equation. This represents a significant failing in what is now a standard test of singularity structure. We present a solution of this problem.     

On the use of the Painlevé property for obtaining miura type transformation

We present a method for obtaining an associated hierarchy of evolution equations possessing the Painlevé property from a given hierarchy which possesses the Painlevé property. This method is applied to the classical Boussinesq hierarchy to obtain the Miura type transformation and the modified classical Boussinesq hierarchy. It is also used to construct a large hierarchy of evolution equations which possess the Painlevé property and include the classical Boussinesq the Jaulent Miodek, the dispersive long wave hierarchy as special cases. All these hierarchies have the same modified hierarchy.The projection supported by the National Natural Science Fundation of China.     

Integrable hierarchies: Painlevé indices and compatibility conditions

We consider the application of the Weiss-Tabor-Carnevale (WTC) Painlevé test to hierarchies of completely integrable evolution equations. A method of constructing the Painlevé index polynomial for such hierarchies is illustrated. For Burgers' hierarchy we are able to show that all WTC compatibility conditions are satisfied. This allows a simple construction of the Painlevé-Bäcklund transformation obtained from truncation of the principal Painlevé expansion.Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 509–516, June, 1994.     

On some aspects of Bäcklund transformations

The Painlevé differential equations (P₂-P₆) possess Bäcklund transformations which relate one solution to another solution either of the same equation, with different values of the parameters, or another such equation. We review a method for deriving difference equations, the discrete Painlevé equations in particular, from Bäcklund transformations of the continuous Painlevé equations. Then, we prove the existence of an algebraic formula relating three inconsecutive solutions of the same Bäcklund hierarchy for P₃ and P₄.     

Value distribution of meromorphic solutions of certain difference Painlevé equations

The Borel exceptional value and the exponents of convergence of poles, zeros and fixed points of finite order transcendental meromorphic solutions for difference Painlevé I and II equations are estimated. And the forms of rational solutions of the difference Painlevé II equation and the autonomous difference Painlevé I equation are also given. It is also proved that the non-autonomous difference Painlevé I equation has no rational solution.     

Symmetry and singularity properties of second-order ordinary differential equations of Lie's type III

We extend the work of Abraham-Shrauner [B. Abraham-Shrauner, Hidden symmetries and linearization of the modified Painlevé-Ince equation, J. Math. Phys. 34 (1993) 4809-4816] on the linearization of the modified Painlevé-Ince equation to a wider class of nonlinear second-order ordinary differential equations invariant under the symmetries of time translation and self-similarity. In the process we demonstrate a remarkable connection with the parameters obtained in the singularity analysis of this class of equations.     

Special functions arising from discrete Painlevé equations: A survey

This article is a survey on recent studies on special solutions of the discrete Painlevé equations, especially on hypergeometric solutions of the q-Painlevé equations. The main part of this survey is based on the joint work [K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Hypergeometric solutions to the q-Painlevé equations, IMRN 2004 47 (2004) 2497–2521, K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Construction of hypergeometric solutions to the q-Painlevé equations, IMRN 2005 24 (2005) 1439–1463] with Kajiwara, Masuda, Ohta and Yamada. After recalling some basic facts concerning Painlevé equations for comparison, we give an overview of the present status of studies on difference (discrete) Painlevé equations as a source of special functions.     

Asymptotic expansions for partial solutions of the sixth Painlevé equation

A formalism for an averaging method for the Painlevé equations, in particular, the sixth equation, is developed. The problem is to describe the asymptotic behavior of the sixth Painlevé transcendental in the case where the module of the independent variable tends to infinity. The corresponding expansions contain an elliptic function (ansatz) in the principal term. The parameters of this function depend on the variable because of the modulation equation. The elliptic ansatz and the modulation equation for the sixth Painlevé equation are obtained in their explicit form. A partial solution of the modulation equation leading to a previously unknown asymptotic expansion for the partial solution of the sixth Painlevé equation is obtained.     

CKP and BKP Equations Related to the Generalized Quartic Hénon–Heiles Hamiltonian

In their classification of soliton equations from a group theoretical standpoint according to the representation of infinite Lie algebras, Jimbo and Miwa listed bilinear equations of low degree for the KP and the modified KP hierarchies. In this list, we consider the (1+1)-dimensional reductions of three particular equations of special interest for establishing some new links with the generalized Hénon–Heiles Hamiltonian, possibly useful for integrating the latter with functions having the Painlevé property. Two of those partial differential equations have N-soliton solutions that, as for the Kaup–Kupershmidt equation, can be written as the logarithmic derivative of a Grammian. Moreover, they can describe head-on collisions of solitary waves of different type and shape.     

KdV Equations and integrability detectors

We present a review of the various integrability detectors that have been developed based on the study of the singularities of the solutions of a given equation: the Painlevé method for continuous systems, and the singularity confinement approach for discrete ones. In each case the KdV equation was instrumental in the formulation of the conjectures relating the singularity structure to integrability.     

On non-isospectral flows,Painlevé equations,and symmetries of differential and difference equations

We identify the Painlevé Lax pairs with those corresponding to stationary solutions of non-isospectral flows, both for partial differential equations and differential-difference equations. We discuss symmetry reductions of integrable differential-difference equations and show that, in contrast with the continuous case, where Painlevé equations naturally arise, in the discrete case the so-called discrete Painlevé equations cannot be obtained in this way. Actually, symmetry reductions of integrable differential-difference equations naturally provide delay Painlevé equations.In Memory of Prof. M. C. PolivanovDipartimento di Fisica, P. le A. Moro 2, 00185 Roma, Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy. Departamento de Fisica Teorica, Facultad de Fisicas, Universidad Complutense, 28040 Madrid, Spain. W561@emducm11.bitnet. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 473–480, December, 1992.     

The discrete Korteweg-de Vries equation

We review the different aspects of integrable discretizations in space and time of the Korteweg-de Vries equation, including Miura transformations to related integrable difference equations, connections to integrable mappings, similarity reductions and discrete versions of Painlevé equations as well as connections to Volterra systems.